Dear MHF members,

my problem reads as follows. I have some ideas about some parts of the problem but I would like to share it completely because I find the problem nice.

Problem. Consider the system $\displaystyle \mathbf{X}^{\prime}=\mathbf{A}\mathbf{X}$, where $\displaystyle \mathbf{A}$ is a $\displaystyle 30\times30$ constant matrix with only two distinct eigenvalues $\displaystyle \lambda_{1}=1$ and $\displaystyle \lambda_{2}=2$ Suppose that the Jordan form $\displaystyle \mathbf{J}$ of $\displaystyle \mathbf{A}$ has Jordan blocks $\displaystyle \mathbf{J}_{3},\mathbf{J}_{3},\mathbf{J}_{6}$ corresponding to $\displaystyle \lambda_{1}$ and blocks $\displaystyle \mathbf{J}_{1},\mathbf{J}_{2},\mathbf{J}_{3},$$\displaystyle \mathbf{J}_{3},\mathbf{J}_{4},\mathbf{J}_{5}$ corresponding to $\displaystyle \lambda_{2}$, where $\displaystyle \mathbf{J}_{k}$ is a $\displaystyle k\times k$ Jordan block.

- How many linearly independent eigenvectors does $\displaystyle \mathbf{A}$ have?
- Suppose that $\displaystyle \mathbf{V}$ is not an eigenvector but satisfies $\displaystyle (\mathbf{A}-\lambda_{2}\mathbf{I})^{3}\mathbf{V}=0$. How many linearly independent solution vectors $\displaystyle \mathbf{V}$ are there?
- Suppose that the Jordan blocks occur down the diagonal of the $\displaystyle \mathbf{J}$ in the order stated above. Which matrix elements of $\displaystyle \lim_{t\to\infty}t^{4}\mathrm{e}^{-\lambda_{2}t}\mathrm{e}^{\mathbf{J}t}$ are non-zero and what are their values? $\displaystyle \rule{0.2cm}{0.2cm}$

Thanks a lot.

bkarpuz